Middle school students' understanding of number sense related to percent

School Science and Mathematics, Jan 1997 by Gay, A Susan, Aichele, Douglas B

This study examined middle school students' understanding of percent, focusing on number sense. Participants in the study were 106 seventh-grade and 93 eighth-grade students. Students were given a written test that included 21 multiple-choice questions and an open-ended item. Research interviews were conducted with 28 selected students. Students performed better interpreting a quantity expressed as a percent given a pictorial continuous region than when a pictorial discrete set of circles was given. Students had difficulty interpreting a quantity expressed as a percent of a number. The strategies used by students to make comparisons about percents represent a wide range of correct and incorrect approaches to the questions. In addition to the use of 50% and 100% as common reference points, students successfully applied fractional relationships, estimation and mental computation to make comparisons. A variety of inappropriate strategies which included computational procedures and numerical comparisons were also employed, some of which resulted in the correct multiple-choice response.

In the Curriculum and Evaluation Standards for School Mathematics, the National Council of Teachers of Mathematics [NCTM] (1989) suggested increased attention in the middle grades on developing an understanding of percent and number sense. However, studies have documented that students do not perform well on questions dealing with percent (Allinger & Payne, 1986; Comstock & Demana,1987; Hart,1981; Kouba, Carpenter, & Swafford, 1989). In particular, students did not perform well on questions which focused on the concept of percent during the National Assessment of Educational Progress [NAEP] conducted in 1986 (Kouba et al., 1989). On a comparison question, such as "76% of 20 is greater than, less than, or equal to 20," only 37% of seventh graders and 69% of eleventh graders responded correctly (Kouba et al., 1989). Results from the National Assessment of Educational Progress conducted in 1992 showed some improvement among eighth-grade students at level 300, which designates the level along NAEP's 0 to 500 point scale where students are expected to reason and estimate with percents (Mullis, Dossey, Owen&Phillips, 1993). It was estimated that 20 percent of eighth-grade students nationally performed at this level or above in 1992, which is an increase from IS percent in 1990 (Mullis et al., 1993).

Number sense is a term which encompasses several skills related to a "common sense" about numbers. Those skills include: (1) having well-understood number meanings; (2) having developed multiple relationships among numbers; (3) recognizing the relative magnitudes of numbers; and (4) knowing the relative effect of operating on numbers (NCTM,1989). Understanding a number as a quantity of a specific magnitude and being able to judge how it compares to another number is basic to number sense (Sowder, 1988). In addition to having number sense about whole numbers, fractions, and decimals, students should develop number sense about percent. This includes understanding the meaning of numbers expressed as percents, developing equivalent expressions for percents, comparing quantities expressed as percents, and recognizing the relative effect of finding a percent of a number.

Research has indicated that percent is a difficult topic in the middle grades' mathematics curriculum (Allinger & Payne, 1986; Hart, 1981; McGivney & Nitschke, 1988; Wiebe,1986). The record of research investigating computation with percents and solving types of percent problems is extensive, though little attention has been given to students' number sense about percent. Work by Kircher (1926), Edwards (1930), and Guiler (1946a, 1946b) focused on students' understanding of percent by studying student errors on percent computation problems. A number of researchers have investigated approaches to teaching the three cases of percent problems (Bidwell, 1969; Kenney & Stockton, 1958; Maxim, 1983; May, 1965; McCarty,1967; McMahon,1959; Montgomery,1958; Tredway & Hollister, 1963; Wynn, 1965). Additional work has been done proposing techniques or procedures to help students work percent problems (Allinger, 1985; Bennett, Jr. & Nelson, 1994; Dewar, 1984; Dollins, 1981; McGivney & Nitschke, 1988; Osiecki, 1988; Wiebe,1986). Recent work (Costa,1994; Lembke & Reys, 1994; Risacher, 1993) has focused on students' understanding of percent and ability to solve percent problems.

The study of percent in the school curriculum is typically concentrated in the middle grades. Students are taught to find equivalent expressions among fractions, decimals, and percents. The emphasis of this work in the curriculum is the three cases of percent problems and some applications of percent. Cases of percent involve the relationship, percentage equals base times rate, where the rate is expressed as a percent (Underhill, 1972). Case one percent problems are characterized by finding the percentage as the product of the base and the rate (Schminke, Maertens, & Arnold, 1973). In case two and case three percent problems, the unknowns are the rate and the base, respectively (Schminke et al., 1973).